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Conventionally, modelling of multivariate extremes has been based on the class of multivariate extreme value distributions. More recently, other classes have been developed, allowing for the possibility that, whilst dependence is observed at finite levels, the limit distribution is independent. A number of articles have shown this development to be important for accurate estimation of the extremal properties, both of theoretical processes and observed datasets. It has also been shown that, so far as dependence is concerned, the choice between modelling with either asymptotically dependent or asymptotically independent distributions can be far more influential than model choice within either of these two classes. In this paper we explore the issue of modelling across both classes, examining in particular the effect of uncertainty caused by lack of knowledge about the status of asymptotic dependence. This is achieved by new multivariate models whose parameter spaces are such that asymptotic dependence occurs on a boundary. Standard techniques in Bayesian inference, implemented through Markov chain Monte Carlo, enable inferences to be drawn that assign posterior probability mass to the boundary region. The techniques are illustrated on a set of oceanographic data for which previous analyses have shown that it is difficult to resolve the question of asymptotic dependence status, which is however important in model extrapolation.